Curvilinear approximation method

Abstract

A method for approximately representing a first curve having anchor points and a control point, wherein the first curve is subdivided into a plurality of segments and an approximate curve is constructed by joining the points corresponding to the ends of the segments with straight lines. The number of segments is determined by sequentially subdividing the first curve until sum of the lengths of lines joining the anchor points and control points is less than a predetermined value.

What we claim is: 
     
       1. An apparatus for constructing a second curve that is an approximation of a first curve, comprising a computer having a program memory, wherein said program memory is connected to control said computer to produce signals corresponding the coordinates of a plurality of subdivisions of said first curve and to generate signals corresponding to lines interconnecting adjacent subdividing points with straight lines, the improvement wherein said program memory is connected to said computer to control said computer to generate a sequence of signals corresponding to coordinates of first and second subdivided curve sections of said first curve; said program memory having means to control said computer in response to said sequence of signals, to determine the difference between the total length of a line segment when respectively connecting points at both ends of said first subdivided curve section to a point other than said points of both ends of said first subdivided curve section with straight line segments and the length of the line segment when connecting said points of both ends thereof with a straight line; said program memory further comprising means responsive to said determination of said difference to control said computer to determine the number of subdivisions of said first curve on the basis of the number of subdividing operations until said difference becomes smaller than a predetermined value, said program memory further comprising means responsive to the determination of said number by said computer for controlling said computer to produce output signals corresponding to coordinates of said number of subdivisions of said first curve, said apparatus further comprising means responsive to said output signals for constructing said approximate curve by producing signals corresponding to straight lines joining points corresponding to the ends of said determined number of subdivisions.   
     
     
       2. An apparatus for constructing a second curve that is an approximation of a first curve, the first curve having two anchor points and at least one control point, comprising a computer having a program memory means, said program memory means being connected to said computer to control said computer to produce signals corresponding the coordinates of a plurality of subdivisions of said first curve and to generate signals corresponding to lines interconnecting adjacent subdividing points with straight lines, said program memory means comprising means for controlling said computer to generate a sequence of signals corresponding to coordinate of first and second subdivided curve sections of said first curve; said program memory means comprising first means responsive to generation of said sequence of signals by said computer to control said computer to determine the difference between the total length of a line segment when respectively connecting points at both ends of said first subdivided curve section to a point other than said points of both ends of said first subdivided curve section with straight line segments and the length of the line segment when connecting said points of both ends thereof with a straight line; said program memory means further comprising second means responsive to said determination of said difference by said computer to control said computer to determine the number of subdivisions of said first curve on the basis of the number of subdividing operations until said difference becomes smaller than a predetermined value, said program memory means further comprising third means responsive to the determination by said computer of said number for controlling said computer to produce output signals corresponding to coordinates of said number of subdivisions of said first curve, and further comprising means responsive to said output signals for constructing said approximate curve by producing signals corresponding to straight lines joining points corresponding to the ends of said determined number of subdivisions. 
     
     
       3. An apparatus for constructing a second curve that is an approximation of a first curve, comprising: a computer having a program memory,   first, second, third and fourth random access memories coupled to said computer,   said first memory having stored therein predetermined values of WA, WB, WC and WD that are cubic functions of a variable t, defining said first curve, for a sequence of a plurality of equally spaced values of t, wherein 0≦t≦1,   said second memory having stored therein, coordinates of anchor points PA and PD and control points PB and PC of a plurality of segments of said first curve,   said third memory having stored therein a predetermined separate step number n for each of said segments, wherein n is different for at least some of said segments,   said program memory comprising means for calculating, for each segment, coordinates thereof in accordance with the relationship:   Q=WA·PA+WB·PB+WC·PC+WD·PC     employing only nth stored values of said first memory and nth coordinates stored in said first and second memories, respectively, and means for storing said calculated coordinates in said fourth memory, said calculated coordinates comprising said adjacent determined coordinate points.